English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This thesis investigates parametrized quantum spin systems in thethermodynamic limit from a $C^*$-algebraic point of view. Our main physicalresult is the construction of a phase invariant for one-dimensional quantumspin chains parametrized by a topological space $X$. This invariant isconstructed using $C^*$-algebraic techniques and takes values in degree one\v{C}ech cohomology $H^1(X;\mathbb{P}\mathrm{U}(\mathscr{H}))$, where$\mathbb{P}\mathrm{U}(\mathscr{H})$ is the projective unitary group of aninfinite-dimensional Hilbert space $\mathscr{H}$, endowed with the strongoperator topology. Using Dixmier-Douady theory [Bry93, DD63, HJJS08] one mayequivalently view this as an element of $H^3(X;\mathbb{Z})$. An exactlysolvable model of a one-dimensional spin system parametrized by the 3-sphere $X= \mathbb{S}^3$ is presented and it is shown that its invariant is nontrivial[WQB+22].  We also prove several mathematical results on topological aspects of the purestate space $\mathscr{P}(\mathfrak{A})$ of a $C^*$-algebra $\mathfrak{A}$. Weprove that $\mathscr{P}(\mathfrak{A})$, endowed with the weak* topology, hastrivial fundamental group for every UHF algebra $\mathfrak{A}$ [BHM+23]. Wereview the results of [SMQ+22] that show how the outputs of the GNSrepresentation and Kadison transitivity theorem can be understood to dependcontinuously on their inputs. These results are expanded upon with theconstruction of a distinguished \v{C}ech class in$H^1(\mathscr{P}(\mathfrak{A});\mathrm{U}(1))$ that generalizes the principal$\mathrm{U}(1)$-bundle $\mathbb{S} \mathscr{H} \rightarrow \mathbb{P}\mathscr{H}$ in a representation independent way, where $\mathbb{S}\mathscr{H}$ and $\mathbb{P} \mathscr{H}$ are the unit sphere and projectiveHilbert space of $\mathscr{H}$. Finally, we prove a selection theorem in thevein of [SMQ+22] for the weak* topology on $\mathscr{P}(\mathfrak{A})$.

A $C^*$-Algebraic Approach to Parametrized Quantum Spin Systems and Their Phases in One Spatial Dimension